In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a triacontagon or 30-gon is a thirty-sided
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
. The sum of any triacontagon's interior angles is
5040 degrees.
Regular triacontagon
The ''
regular triacontagon'' is a
constructible polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...
, by an edge-
bisection of a regular
pentadecagon
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
Regular pentadecagon
A ''regular polygon, regular pentadecagon'' is represented by Schläfli symbol .
A Regular polygon, regular pentadecagon has interior angl ...
, and can also be constructed as a
truncated pentadecagon
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
Regular pentadecagon
A ''regular polygon, regular pentadecagon'' is represented by Schläfli symbol .
A Regular polygon, regular pentadecagon has interior angl ...
, t. A
truncated triacontagon, t, is a
hexacontagon, .
One interior angle in a
regular triacontagon is 168 degrees, meaning that one exterior angle would be 12°. The triacontagon is the largest regular polygon whose interior angle is the sum of the
interior angles of smaller
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
s: 168° is the sum of the interior angles of the
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
(60°) and the
regular pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
(108°).
The
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
of a regular triacontagon is (with )
:
The
inradius of a regular triacontagon is
:
The
circumradius of a regular triacontagon is
:
Construction
As 30 = 2 × 3 × 5, a regular triacontagon is
constructible using a
compass and straightedge.
Symmetry
The ''regular triacontagon'' has Dih
30 dihedral symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...
, order 60, represented by 30 lines of reflection. Dih
30 has 7 dihedral subgroups: Dih
15, (Dih
10, Dih
5), (Dih
6, Dih
3), and (Dih
2, Dih
1). It also has eight more
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
symmetries as subgroups: (Z
30, Z
15), (Z
10, Z
5), (Z
6, Z
3), and (Z
2, Z
1), with Z
n representing π/''n'' radian rotational symmetry.
John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.
These lower symmetries allows degrees of freedoms in defining irregular triacontagons. Only the g30 subgroup has no degrees of freedom but can seen as
directed edge
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pai ...
s.
Dissection
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
states that every
zonogon
In geometry, a zonogon is a centrally-symmetric, convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations.
Examples
A regular polygon is a zonogon if and ...
(a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms.
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the ''regular triacontagon'', ''m''=15, it can be divided into 105: 7 sets of 15 rhombs. This decomposition is based on a
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
projection of a
15-cube.
Triacontagram
A triacontagram is a 30-sided
star polygon. There are 3 regular forms given by
Schläfli symbols , , and , and 11 compound star figures with the same
.
There are also
isogonal triacontagrams constructed as deeper truncations of the regular
pentadecagon
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
Regular pentadecagon
A ''regular polygon, regular pentadecagon'' is represented by Schläfli symbol .
A Regular polygon, regular pentadecagon has interior angl ...
and pentadecagram , and inverted pentadecagrams , and . Other truncations form double coverings: t2, t2, t2, and t2.
[The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', ]Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent[Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...](_blank)
for three 8-dimensional polytopes with E8 symmetry, shown in orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
s in the E8 Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ar ...
. It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H4 Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ar ...
.
The regular triacontagram is also the Petrie polygon for the great grand stellated 120-cell and grand 600-cell.
References
Naming Polygons and Polyhedra
triacontagon
{{polygons
Constructible polygons
Polygons by the number of sides